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Andrzej Lingas

Publications and source records attributed to Andrzej Lingas.

At least 19 recordsLinked to original sources
arXiv2026

On Solving Problems of Substantially Super-linear Complexity in $N^{o(1)}$ Rounds in the MPC Model

We study the possibility of designing $N^{o(1)}$-round protocols for problems of substantially super-linear polynomial-time (sequential) complexity in the model of Massively Parallel Computation, where $N$ is the input size. We show that if the machines are not equipped with relatively large local memory and their number does not exceed $N$, then the exponent of the average time complexity of the local computation performed by a machine in a round (in terms of local memory size) in such protocols must be larger than the exponent of the time complexity of the given problem.

cs.DC↗
arXiv2025

Fast approximate $\ell$-center clustering in high dimensional spaces

We study the design of efficient approximation algorithms for the $\ell$-center clustering and minimum-diameter $\ell$-clustering problems in high dimensional Euclidean and Hamming spaces. Our main tool is randomized dimension reduction. First, we present a general method of reducing the dependency of the running time of a hypothetical algorithm for the $\ell$-center problem in a high dimensional Euclidean space on the dimension size. Utilizing in part this method, we provide $(2+\epsilon)$- approximation algorithms for the $\ell$-center clustering and minimum-diameter $\ell$-clustering problems in Euclidean and Hamming spaces that are substantially faster than the known $2$-approximation ones when both $\ell$ and the dimension are super-logarithmic. Next, we apply the general method to the recent fast approximation algorithms with higher approximation guarantees for the $\ell$-center clustering problem in a high dimensional Euclidean space. Finally, we provide a speed-up of the known $O(1)$-approximation method for the generalization of the $\ell$-center clustering problem to include $z$ outliers (i.e., $z$ input points can be ignored while computing the maximum distance of an input point to a center) in high dimensional Euclidean and Hamming spaces.

cs.DS↗
arXiv2025

Approximate all-pairs Hamming distances and 0-1 matrix multiplication

Arslan showed that computing all-pairs Hamming distances is easily reducible to arithmetic 0-1 matrix multiplication (IPL 2018). We provide a reverse, linear-time reduction of arithmetic 0-1 matrix multiplication to computing all-pairs distances in a Hamming space. On the other hand, we present a fast randomized algorithm for approximate all-pairs distances in a Hamming space. By combining it with our reduction, we obtain also a fast randomized algorithm for approximate 0-1 matrix multiplication. Next, we present an output-sensitive randomized algorithm for a minimum spanning tree of a set of points in a generalized Hamming space, the lower is the cost of the minimum spanning tree the faster is our algorithm. Finally, we provide $(2+\epsilon)$- approximation algorithms for the $\ell$-center clustering and minimum-diameter $\ell$-clustering problems in a Hamming space $\{0,1\}^d$ that are substantially faster than the known $2$-approximation ones when both $\ell$ and $d$ are super-logarithmic.

cs.DS↗
arXiv2025

Multiplication of 0-1 matrices via clustering

We study applications of clustering (in particular, the $k$-center clustering problem) in the design of efficient and practical algorithms for computing an approximate and the exact arithmetic matrix product of two 0-1 rectangular matrices with clustered rows or columns, respectively. Our results in part can be regarded as an extension of the clustering-based approach to Boolean square matrix multiplication due to Arslan and Chidri (CSC 2011). First, we provide a simple and efficient deterministic algorithm for approximate matrix product of 0-1 matrices, where the additive error is proportional to the minimum maximum radius in an $\ell$-center clustering of the rows of the first matrix or an $k$-center clustering of the columns of the second matrix. Next, we use the approximation algorithm as a preprocessing after which a query asking for the exact value of an arbitrary entry in the product matrix can be answered in time proportional to the additive error. As a consequence, we obtain a simple deterministic algorithm for the exact matrix product of 0-1 matrices. We also present an improved simple deterministic algorithm for the exact product and in addition, faster analogous randomized algorithms for an approximate and the exact matrix products of 0-1 matrices based on randomized $\ell$ and $k$-center clustering.

cs.DS↗
arXiv2024

Boolean Matrix Multiplication for Highly Clustered Data on the Congested Clique

We present a protocol for the Boolean matrix product of two $n\times b$ Boolean matrices on the congested clique designed for the situation when the rows of the first matrix or the columns of the second matrix are highly clustered in the space $\{0,1\}^n.$ With high probability (w.h.p), it uses $\tilde{O}\left(\sqrt {\frac M n+1}\right)$ rounds on the congested clique with $n$ nodes, where $M$ is the minimum of the cost of a minimum spanning tree (MST) of the rows of the first input matrix and the cost of an MST of the columns of the second input matrix in the Hamming space $\{0,1\}^n.$ A key step in our protocol is the computation of an approximate minimum spanning tree of a set of $n$ points in the space $\{0,1\}^n$. We provide a protocol for this problem (of interest in its own rights) based on a known randomized technique of dimension reduction in Hamming spaces. W.h.p., it constructs an $O(1)$-factor approximation of an MST of $n$ points in the Hamming space $\{ 0,\ 1\}^n$ using $O(\log^3 n)$ rounds on the congested clique with $n$ nodes.

cs.DS↗
arXiv2024

A Note on Solving Problems of Substantially Super-linear Complexity in $N^{o(1)}$ Rounds of the Congested Clique

We study the possibility of designing $N^{o(1)}$-round protocols for problems of substantially super-linear polynomial-time (sequential) complexity on the congested clique with about $N^{1/2}$ nodes, where $N$ is the input size. We show that the average time complexity of the local computation performed at a clique node (in terms of the size of the data received by the node) in such protocols has to be substantially larger than the time complexity of the given problem.

cs.DC↗
arXiv2024

The Voronoi Diagram of Weakly Smooth Planar Point Sets in $O(\log n)$ Deterministic Rounds on the Congested Clique

We study the problem of computing the Voronoi diagram of a set of $n^2$ points with $O(\log n)$-bit coordinates in the Euclidean plane in a substantially sublinear in $n$ number of rounds in the congested clique model with $n$ nodes. Recently, Jansson et al. have shown that if the points are uniformly at random distributed in a unit square then their Voronoi diagram within the square can be computed in $O(1)$ rounds with high probability (w.h.p.). We show that if a very weak smoothness condition is satisfied by an input set of $n^2$ points with $O(\log n)$-bit coordinates in the unit square then the Voronoi diagram of the point set within the unit square can be computed in $O(\log n)$ rounds in this model.

cs.CG↗
arXiv2023

$(\min,+)$ Matrix and Vector Products for Inputs Decomposable into Few Monotone Subsequences

We study the time complexity of computing the $(\min,+)$ matrix product of two $n\times n$ integer matrices in terms of $n$ and the number of monotone subsequences the rows of the first matrix and the columns of the second matrix can be decomposed into. In particular, we show that if each row of the first matrix can be decomposed into at most $m_1$ monotone subsequences and each column of the second matrix can be decomposed into at most $m_2$ monotone subsequences such that all the subsequences are non-decreasing or all of them are non-increasing then the $(\min,+)$ product of the matrices can be computed in $O(m_1m_2n^{2.569})$ time. On the other hand, we observe that if all the rows of the first matrix are non-decreasing and all columns of the second matrix are non-increasing or {\em vice versa} then this case is as hard as the general one. Similarly, we also study the time complexity of computing the $(\min,+)$ convolution of two $n$-dimensional integer vectors in terms of $n$ and the number of monotone subsequences the two vectors can be decomposed into. We show that if the first vector can be decomposed into at most $m_1$ monotone subsequences and the second vector can be decomposed into at most $m_2$ subsequences such that all the subsequences of the first vector are non-decreasing and all the subsequences of the second vector are non-increasing or {\em vice versa} then their $(\min,+)$ convolution can be computed in $\tilde{O}(m_1m_2n^{1.5})$ time. On the other, the case when both vectors are non-decreasing or both of them are non-increasing is as hard as the general case.

cs.DS↗
arXiv2023

Finding Small Complete Subgraphs Efficiently

(I) We revisit the algorithmic problem of finding all triangles in a graph $G=(V,E)$ with $n$ vertices and $m$ edges. According to a result of Chiba and Nishizeki (1985), this task can be achieved by a combinatorial algorithm running in $O(m \alpha) = O(m^{3/2})$ time, where $\alpha= \alpha(G)$ is the graph arboricity. We provide a new very simple combinatorial algorithm for finding all triangles in a graph and show that is amenable to the same running time analysis. We derive these worst-case bounds from first principles and with very simple proofs that do not rely on classic results due to Nash-Williams from the 1960s. Our experimental results show that our simple algorithm for triangle listing is substantially faster in practice than that of Chiba and Nishizeki on all examples of real-world graphs we tried. (II) We extend our arguments to the problem of finding all small complete subgraphs of a given fixed size. We show that the dependency on $m$ and $\alpha$ in the running time $O(\alpha^{\ell-2} \cdot m)$ of the algorithm of Chiba and Nishizeki for listing all copies of $K_\ell$, where $\ell \geq 3$, is asymptotically tight. (III) We give improved arboricity-sensitive running times for counting and/or detection of copies of $K_\ell$, for small $\ell \geq 4$. A key ingredient in our algorithms is, once again, the algorithm of Chiba and Nishizeki. Our new algorithms are faster than all previous algorithms in certain high-range arboricity intervals for every $\ell \geq 7$.

cs.DS↗
arXiv2023

Convex Hulls, Triangulations, and Voronoi Diagrams of Planar Point Sets on the Congested Clique

We consider geometric problems on planar $n^2$-point sets in the congested clique model. Initially, each node in the $n$-clique network holds a batch of $n$ distinct points in the Euclidean plane given by $O(\log n)$-bit coordinates. In each round, each node can send a distinct $O(\log n)$-bit message to each other node in the clique and perform unlimited local computations. We show that the convex hull of the input $n^2$-point set can be constructed in $O(\min\{ h,\log n\})$ rounds, where $h$ is the size of the hull, on the congested clique. We also show that a triangulation of the input $n^2$-point set can be constructed in $O(\log^2n)$ rounds on the congested clique. Finally, we demonstrate that the Voronoi diagram of $n^2$ points with $O(\log n)$-bit coordinates drawn uniformly at random from a unit square can be computed within the square with high probability in $O(1)$ rounds on the congested clique.

cs.DC↗
arXiv2023

Improved Lower Bounds for Monotone q-Multilinear Boolean Circuits

A monotone Boolean circuit is composed of OR gates, AND gates and input gates corresponding to the input variables and the Boolean constants. It is $q$-multilinear if for each its output gate $o$ and for each prime implicant $s$ of the function computed at $o$, the arithmetic version of the circuit resulting from the replacement of OR and AND gates by addition and multiplication gates, respectively, computes a polynomial at $o$ which contains a monomial including the same variables as $s$ and each of the variables in $s$ has degree at most $q$ in the monomial. First, we study the complexity of computing semi-disjoint bilinear Boolean forms in terms of the size of monotone $q$-multilinear Boolean circuits. In particular, we show that any monotone $1$-multilinear Boolean circuit computing a semi-disjoint Boolean form with $p$ prime implicants includes at least $p$ AND gates. We also show that any monotone $q$-multilinear Boolean circuit computing a semi-disjoint Boolean form with $p$ prime implicants has $\Omega(\frac p {q^4})$ size. Next, we study the complexity of the monotone Boolean function $Isol_{k,n}$ that verifies if a $k$-dimensional Boolean matrix has at least one $1$ in each line (e.g., each row and column when $k=2$), in terms of monotone $q$-multilinear Boolean circuits. We show that that any $\Sigma_3$ monotone Boolean circuit for $Isol_{k,n}$ has an exponential in $n$ size or it is not $(k-1)$-multilinear.

cs.CC↗
arXiv2022

Perpetual maintenance of machines with different urgency requirements

A garden $G$ is populated by $n\ge 1$ bamboos $b_1, b_2, ..., b_n$ with the respective daily growth rates $h_1 \ge h_2 \ge \dots \ge h_n$. It is assumed that the initial heights of bamboos are zero. The robotic gardener maintaining the garden regularly attends bamboos and trims them to height zero according to some schedule. The Bamboo Garden Trimming Problem (BGT) is to design a perpetual schedule of cuts to maintain the elevation of the bamboo garden as low as possible. The bamboo garden is a metaphor for a collection of machines which have to be serviced, with different frequencies, by a robot which can service only one machine at a time. The objective is to design a perpetual schedule of servicing which minimizes the maximum (weighted) waiting time for servicing. We consider two variants of BGT. In discrete BGT the robot trims only one bamboo at the end of each day. In continuous BGT the bamboos can be cut at any time, however, the robot needs time to move from one bamboo to the next. For discrete BGT, we show tighter approximation algorithms for the case when the growth rates are balanced and for the general case. The former algorithm settles one of the conjectures about the Pinwheel problem. The general approximation algorithm improves on the previous best approximation ratio. For continuous BGT, we propose approximation algorithms which achieve approximation ratios $O(\log \lceil h_1/h_n\rceil)$ and $O(\log n)$.

cs.DS↗
arXiv2021

Breaking the hegemony of the triangle method in clique detection

We consider the fundamental problem of detecting/counting copies of a fixed pattern graph in a host graph. The recent progress on this problem has not included complete pattern graphs, i.e., cliques (and their complements, i.e., edge-free pattern graphs, in the induced setting). The fastest algorithms for the aforementioned patterns are based on a straightforward reduction to triangle detection/counting. We provide an alternative method of detection/counting copies of fixed size cliques based on a multi-dimensional matrix product. It is at least as time efficient as the triangle method in cases of $K_4$ and $K_5.$ The complexity of the multi-dimensional matrix product is of interest in its own rights. We provide also another alternative method for detection/counting $K_r$ copies, again time efficient for $r\in \{ 4,\ 5 \}$.

cs.DS↗
arXiv2021

On Truly Parallel Time in Population Protocols

The {\em parallel time} of a population protocol is defined as the average number of required interactions that an agent in the protocol participates, i.e., the quotient between the total number of interactions required by the protocol and the total number $n$ of agents, or just roughly the number of required rounds with $n$ interactions. This naming triggers an intuition that at least on the average a round of $n$ interactions can be implemented in $O(1)$ parallel steps. We show that when the transition function of a population protocol is treated as a black box then the expected maximum number of parallel steps necessary to implement a round of $n$ interactions is $\Omega (\frac {\log n}{\log \log n})$. We also provide a combinatorial argument for a matching upper bound on the number of parallel steps in the average case under additional assumptions.

cs.DC↗
arXiv2021

An output-sensitive algorithm for all-pairs shortest paths in directed acyclic graphs

A straightforward dynamic programming method for the single-source shortest paths problem (SSSP) in an edge-weighted directed acyclic graph (DAG) processes the vertices in a topologically sorted order. First, we similarly iterate this method alternatively in a breadth-first search sorted order and the reverse order on an input directed graph with both positive and negative real edge weights, $n$ vertices and $m$ edges. For a positive integer $t,$ after $O(t)$ iterations in $O(tm)$ time, we obtain for each vertex $v$ a path distance from the source to $v$ not exceeding that yielded by the shortest path from the source to $v$ among the so called {\em$ t+$light paths}. A directed path between two vertices is $t+$light if it contains at most $t$ more edges than the minimum edge-cardinality directed path between these vertices. After $O(n)$ iterations, we obtain an $O(nm)$-time solution to SSSP in directed graphs with real edge weights matching that of Bellman and Ford. Our main result is an output-sensitive algorithm for the all-pairs shortest paths problem (APSP) in DAGs with positive and negative real edge weights. It runs in time $O(\min \{n^{\omega}, nm+n^2\log n\}+\sum_{v\in V}\text{indeg}(v)|\text{leaf}(T_v)|),$ where $n$ is the number of vertices, $m$ is the number of edges, $\omega$ is the exponent of fast matrix multiplication, $\text{indeg}(v)$ stands for the indegree of $v,$ $T_v$ is a tree of lexicographically-first shortest directed paths from all ancestors of $v$ to $v$, and $\text{leaf}(T_v)$ is the set of leaves in $T_v.$ Finally, we discuss an extension of hypothetical improved upper time-bounds for APSP in non-negatively edge-weighted DAGs to include directed graphs with a polynomial number of large directed cycles.

cs.DS↗
arXiv2021

Efficient Assignment of Identities in Anonymous Populations

We consider the fundamental problem of assigning distinct labels to agents in the probabilistic model of population protocols. Our protocols operate under the assumption that the size $n$ of the population is embedded in the transition function. Our labeling protocols are silent w.h.p., i.e., eventually each agent reaches its final state and remains in it forever w.h.p., as well as safe, i.e., never update the label assigned to any single agent. We first present a fast, silent w.h.p.and safe labeling protocol for which the required number of interactions is asymptotically optimal, i.e., $O(n \log n/\epsilon)$ w.h.p. It uses $(2+\epsilon)n+O(n^c)$ states, for any $c<1,$ and the label range $1,\dots,(1+\epsilon)n.$ Furthermore, we consider the so-called pool labeling protocols that include our fast protocol. We show that the expected number of interactions required by any pool protocol is $\ge \frac{n^2}{r+1}$, when the labels range is $1,\dots, n+r<2n.$ Next, we provide a protocol which is silent and safe once a unique leader is provided, and uses only $n+5\sqrt n +O(n^c)$ states, for any $c<1,$ and draws labels from the range $1,\dots,n.$ The expected number of interactions required by the protocol is $O(n^3).$ On the other hand, we show that (even if a unique leader is given in advance) any silent protocol that produces a valid labeling and is safe with probability $>1-\frac 1n$, uses $\ge n+\sqrt {n-1} -1$ states. Hence, our protocol is almost state-optimal. We also present a generalization of the protocol to include a trade-off between the number of states and the expected number of interactions. Furthermore, we show that for any silent and safe labeling protocol utilizing $n+t<2n$ states the expected number of interactions required to achieve a valid labeling is $\ge \frac{n^2}{t+1}$.

cs.DC↗
arXiv2020

Quantum and approximation algorithms for maximum witnesses of Boolean matrix products

The problem of finding maximum (or minimum) witnesses of the Boolean product of two Boolean matrices (MW for short) has a number of important applications, in particular the all-pairs lowest common ancestor (LCA) problem in directed acyclic graphs (dags). The best known upper time-bound on the MW problem for n\times n Boolean matrices of the form O(n^{2.575}) has not been substantially improved since 2006. In order to obtain faster algorithms for this problem, we study quantum algorithms for MW and approximation algorithms for MW (in the standard computational model). Some of our quantum algorithms are input or output sensitive. Our fastest quantum algorithm for the MW problem, and consequently for the related problems, runs in time \tilde{O}(n^{2+\lambda/2})=\tilde{O}(n^{2.434}), where \lambda satisfies the equation \omega(1, \lambda, 1) = 1 + 1.5 \, \lambda and \omega(1, \lambda, 1) is the exponent of the multiplication of an n \times n^{\lambda}$ matrix by an n^{\lambda} \times n matrix. Next, we consider a relaxed version of the MW problem (in the standard model) asking for reporting a witness of bounded rank (the maximum witness has rank 1) for each non-zero entry of the matrix product. First, by adapting the fastest known algorithm for maximum witnesses, we obtain an algorithm for the relaxed problem that reports for each non-zero entry of the product matrix a witness of rank at most \ell in time \tilde{O}((n/\ell)n^{\omega(1,\log_n \ell,1)}). Then, by reducing the relaxed problem to the so called k-witness problem, we provide an algorithm that reports for each non-zero entry C[i,j] of the product matrix C a witness of rank O(\lceil W_C(i,j)/k\rceil ), where W_C(i,j) is the number of witnesses for C[i,j], with high probability. The algorithm runs in \tilde{O}(n^{\omega}k^{0.4653} +n^2k) time, where \omega=\omega(1,1,1).

cs.DS↗